Fast Geometric Point Labeling using Conditional Random Fields
Radu Bogdan Rusu, Andreas Holzbach, Nico Blodow, Michael Beetz
Intelligent Autonomous Systems, Computer Science Department, Technische Universität München
Boltzmannstr. 3, Garching bei München, 85748, Germany
{rusu,holzbach,blodow,beetz}@cs.tum.edu
Abstract— In this paper we present a new approach for
labeling 3D points with different geometric surface primitives
using a novel feature descriptor – the Fast Point Feature
Histograms, and discriminative graphical models. To build
informative and robust 3D feature point representations, our
descriptors encode the underlying surface geometry around a
point p using multi-value histograms. This highly dimensional
feature space copes well with noisy sensor data and is not
dependent on pose or sampling density. By defining classes of 3D
geometric surfaces and making use of contextual information
using Conditional Random Fields (CRFs), our system is able
to successfully segment and label 3D point clouds, based on
the type of surfaces the points are lying on. We validate and
demonstrate the method’s efficiency by comparing it against
similar initiatives as well as present results for table setting
datasets acquired in indoor environments.
based on geometric reasoning. The reason is that while linear
models such as planes can be successfully found, fitting more
complicated geometric primitives like cones becomes very
difficult, due to noise, occlusions, and irregular density, but
also due to the higher number of shape parameters that need
to be found (increased complexity). To deal with such large
solution spaces, heuristic hypotheses generators, while being
unable to provide guarantees regarding the global optimum,
can provide candidates which drastically reduce the number
of models that need to be verified.
I. I NTRODUCTION
Segmenting and interpreting the surrounding environment
that a personal robot operates in from sensed 3D data is
an important research topic. Besides recognizing a certain
location on the map for localization or map refinement
purposes, obtaining accurate and informative object models
is essential for precise manipulation and grasping. Additionally, although the acquired data is discrete and represents
only a few samples of the underlying scanned surfaces, it
quickly becomes expensive to store and work with. It is
therefore imperative to find ways to address this dimensionality problem and group clusters of points sampled from
surfaces with similar geometrical properties together, or in
some sense try to “classify the world”. Achieving the latter
annotates sensed 3D points and geometric structures with
higher level semantics and will greatly simplify research in
the aforementioned topics (e.g. manipulation and grasping).
In general, at the 3D point level, there are two basic
alternatives for acquiring these annotations:
1) use a powerful, discriminative 3D feature descriptor,
and learn different classes of surface or object types
either by supervised or unsupervised learning, and then
use the resultant model to classify newly acquired data;
2) use geometric reasoning techniques such as point cloud
segmentation and region growing, in combination with
robust estimators and non-linear optimization techniques to fit linear (e.g. planes, lines) and non-linear
(e.g. cylinders, spheres) models to the data.
Both approaches have their advantages and disadvantages,
but in most situations machine learning techniques – and
thus the first approach, will outperform techniques purely
Fig. 1.
An ideal 3D point based classification system providing two
different point labels: geometry (L1) and object class (L2).
Additionally, we define a system able to provide two
different hierarchical point annotation levels as an ideal
point classification system. Because labeling objects lying
on a table just with a class label, say mug versus cereal
box, is not enough for manipulation and grasping (because
different mugs have different shapes and sizes and need to be
approached differently by the grasp planner), we require the
classification system to annotate the local geometry around
each point with classes of 3D geometric primitive shapes.
This ensures that besides the object class, we are able to
reconstruct the original object geometry and also plan better
grasping points at the same time. Figure 1 presents the two
required classification levels for 3 points sampled on separate
objects with different local geometry.
This paper concentrates on the first annotation approach,
and introduces a novel 3D feature descriptor for point clouds:
the Fast Point Feature Histograms (FPFH). Our descriptors
are based on the recently proposed Point Feature Histograms
(PFH) [1] and robustly encode the local surface geometry
around a point p using multi-value histograms, but reduce
the original PFH computational complexity from O(N 2 ) to
O(N ) while retaining most of their discriminative power.
The FPFH descriptors cope well with noisy sensor data
acquired from laser sensors and are independent of pose or
sampling density. We present an in-depth analysis on their usage for accurate and reliable point cloud classification, which
provides candidate region clusters for further parameterized
geometric primitive fitting.
By defining classes of 3D geometric surfaces, such as:
planes, edges, corners, cylinders, spheres, etc, and making
use of contextual information using discriminative undirected
graphical models (Conditional Random Fields), our system
is able to successfully segment and label 3D points based
on the type of surfaces the points are lying on. We view
this is as an important building block which speeds up and
improves the recognition and segmentation of objects in real
world scenes.
As an application scenario, we will demonstrate the parameterization and usage of our system for the purpose of
point classification for objects lying on tables in indoor
environments. Figure 2 presents a classification snapshot
for a scanned dataset representing a table with objects in
a kitchen environment. The system outputs point labels for
all objects found in the scene.
Fig. 2. Top: raw point cloud dataset acquired using the platform in Figure 7
containing approximatively 22000 points; bottom: classification results for
points representing the table and the objects on it. The color legend is:
dark green for planar, mid green for convex cylinders, yellow for concave
cylinders, light green for corners, pink for convex edges and brown for
convex tori. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
The remainder of the paper is organized as follows. We
address related work on similar initiatives in the next section,
and describe the Fast Point Feature Histograms (FPFH) in
Section III. The classes of 3D primitive geometric shapes
used for learning the model are presented and analyzed in
Section IV. We introduce our variant of Conditional Random
Fields in Section V and discuss experimental results in
Section VI. Finally we conclude and give insight into our
future work in Section VII.
II. R ELATED W ORK
The area of 3D point-based surface classification has seen
significant progress over time, but more in recent years due
to advances in electronics which fostered both better sensing
devices as well as personal computers able to process the data
faster. Suddenly algorithms for 3D feature estimation which
were intractable for real world applications a few years ago
became viable solutions for certain well-defined problems.
Even so however, most of the work in 3D feature estimation
has been done by the computer graphics community, on
datasets usually acquired from high-end range scanners.
Two widely used geometric point features are the underlying surface’s estimated curvature and normal [2], which
have been also investigated for noisier point datasets [3].
Other proposals for point features include moment invariants
and spherical harmonic invariants [4], and integral volume
descriptors [5]. They have been successfully used in applications such as point cloud registration, and are known
to be translation and rotation invariant, but they are not
discriminative enough for determining the underlying surface
type. Specialized point features that detect peaks, pits and
saddle regions from Gaussian curvature thresholding for the
problem of face detection and pose estimation are presented
in [6]. Conformal factors are introduced by [7], as a measure
of the total curvature of a shape at a given triangle vertex.
In general, descriptors that characterize points with a
single value are not expressive enough to make the necessary
distinctions for point-surface classification. As a direct consequence, most scenes will contain many points with the same
or very similar feature values, thus reducing their informative
characteristics. Alternatively, multiple-value point features
such as curvature maps [8], or spin images [9], are some
of the better local characterizations proposed for 3D meshes
which got adopted for point cloud data. However, these
representations require densely sampled data and are not able
to deal with the amount of noise usually present in 2.5D
scans.
Following the work done by the computer vision community which developed reliable 2D feature descriptors, some
preliminary work has been done recently in extending these
descriptors to the 3D domain. In [10], a keypoint detector
called THRIFT, based on 3D extensions of SURF [11] and
SIFT [12], is used to detect repeated 3D structure in range
data of building facades, however the authors do not address
computational or scalability issues, and fail to compare their
results with the state-of-the art. In [13], the authors propose
RIFT, a rotation invariant 3D feature descriptor utilizing
techniques from SIFT, and perform a comparison of their
work with spin images, showing promising results for the
problem of identifying corresponding points in 3D datasets.
We extend our previous work presented in [1] by computing fast local point feature histograms for each point
in the cloud. We make an in-depth analysis of the points’
signatures for different geometric primitives (i.e. planes,
cylinders, toruses, etc) and show classification results using
two different machine learning methods: Support Vector
Machines (SVM) and Conditional Random Fields (CRF).
The work in [14], [15] is similar to our labeling approach,
as they too use probabilistic graphical models for the problem
of point-cloud classification. In both, sets of simple pointbased 3D features are defined, and a model is trained to
classify points with respect to object classes such as: chairs,
tables, screens, fans, and trash cans [15], respectively: wires,
poles, ground, and scatter [14]. The main difference of our
framework is that we do not craft sets of simpler 3D features
for individual problems to solve, but propose the usage of a
single well defined 3D descriptor for all.
Then, for all sets of computed < α, φ, θ > values in P k ,
a multi-value histogram is created as described in [1]. We
call this the SPFH (Simplified Point Feature Histogram) as
it only describes the direct relationships between a query
point pi and its k neighbors. To capture a more detailed
representation of the underlying surface geometry at P k , in
a second step we refine the final histogram (FPFH) of pi by
weighting all the neighboring SPFH values:
III. FAST P OINT F EATURE H ISTOGRAMS (FPFH)
The computation of a Fast Point Feature Histogram
(FPFH) at a point p relies on the point’s 3D coordinates
and the estimated surface normal at that point. As shown
by [1], the PFH formulation is sufficient for capturing the
local geometry around the point of interest, but if needed, is
extensible to the use of other local properties such as color,
curvature estimates, 2nd order moment invariants, etc.
The main purpose of the FPFH formulation is to create
a feature space in which 3D points lying on primitive
geometric surfaces (e.g. planes, cylinders, spheres, etc) can
be easily identified and labeled. As a requirement then, the
discriminating power of the feature space has to be high
enough so that points on the same surface type can be
grouped in the same class, while points sampled on different
surface types are assigned to different classes. In addition,
the proposed feature space needs to be invariant to rigid
transformations, and insensitive to point cloud density and
noise to a certain degree.
To formulate the FPFH computational model, we introduce
the following notations:
• pi is a 3D point with {xi , yi , zi } geometric coordinates;
• ni is a surface normal estimate at point pi having a
{nxi , nyi , nzi } direction;
• Pn = {p1 , p2 , · · · } is a set of nD points (also represented by P for simplicity);
k
• P is the set of points pj , (j ≤ k), located in the kneighborhood of a query point pi ;
• k · kx is the Lx norm (e.g. k · k1 is the Manhattan or L1
norm, k · k2 is the Euclidean or L2 norm).
Using the above formulation, the estimation of a FPFH
descriptor includes the following steps: i) for each point pi ,
the set of P k neighbors enclosed in the sphere with a given
radius r are selected (k-neighborhood); ii) for every pair of
points pi and pj (i 6= j) in P k , and their estimated normals
ni and nj (pi being the point with a smaller angle between
its associated normal and the line connecting the points), we
define a Darboux uvn frame (u = ni , v = (pj − pi ) ×
u, w = u × v) and compute the angular variations of ni and
nj as follows:
where the weight ωk is chosen as the distance between the
query point pi and a neighboring point pk in some metric
space (Euclidean in our formulation), but could just as well
be selected as a different measure if necessary. To understand
the importance of this weighting scheme, Figure 3 presents
the influence region diagram for a P k set centered at pq . The
SPFH connections are illustrated using red lines, linking pi
with its direct neighbors, while the extra FPFH connections
(due to the additional weighting scheme) are shown in black.
α = v · nj
φ = (u · (pj − pi ))/||pj − pi ||2
θ = arctan(w · nj , u · nj )
(1)
F P F H(pi ) = SP F H(pi ) +
p12
p13
p9
p15
p16
pk2
pq
p10
p14
p17
pk1
pk5
p11
k
1X 1
· SP F H(pk ) (2)
k i=1 ωk
pk3
pk4
p8
p6
p7
Fig. 3. The influence region diagram for a Fast Point Feature Histogram.
Each query point (red) is connected only to its direct k-neighbors (enclosed
by the gray circle). Each direct neighbor is connected to its own neighbors
and the resulted histograms are weighted together with the histogram of the
query point to form the FPFH. The thicker connections contribute to the
FPFH twice.
The number of histogram bins that can be formed in a fully
correlated FPFH space is given by q d , where q is the number
of quantums (i.e. subdivision intervals in a feature’s value
range) and d the number of features selected (e.g. in our
case dividing each feature interval into 5 values will result
in 53 = 125 bins). In this space, a histogram bin increment
corresponds to a point having certain values for all its 3
features. Because of this however, the resulting histogram
will contain a lot of zero bins, and can thus contribute to a
certain degree of information redundancy. A simplification
of the above is to decorrelate the values, that is to simply
create d separate histograms, one for each feature dimension
in Equation 1, and then concatenate them together to create
the FPFH.
In contrast to our PFH formulation [1], the FPFH descriptors reduce the computational complexity from O(N 2 )
to O(N ). In the next section we analyze if the FPFH
descriptors are informative enough to differentiate between
points sampled on different 3D geometric shapes.
IV. 3D G EOMETRIC P RIMITIVES AS C LASS L ABELS
A straightforward analysis of the FPFH discriminating
power can be performed by looking at how features computed for different geometric surfaces resemble or differ from
each other. Since the FPFH computation includes estimated
surface normals, the features can be separately computed and
grouped for both two cases of convex and concave shapes.
Figure 4 presents examples of FPFH signatures for points
lying on 5 different convex surfaces, namely a cylinder, an
edge, a corner, a torus, and finally a plane. To illustrate that
the features are discriminative, in the left part of the figure we
assembled a confusion matrix with gray values representing
the distances between the mean histograms of the different
shapes (a low distance is represented with white, while a
high distance with black), obtained using the Histogram
Intersection Kernel [16]:
d(F P F H1 , F P F H2 ) =
N
X
Q
1
c∈C ψc (xc , yc )
Z
P
Q
1
0
y0
c∈C ψc (xc , yc )
Z
(4)
Therefore we can formulate a CRF model as:
1 Y
p(y|x) =
ψc (xc , yc )
(5)
Z(x)
c∈C
P Q
where Z(x) = y0 c∈C ψc (xc , y 0 ). The factors ψc are
potential functions of the random variables vC within a
clique c ∈ C [18].
According to this mathematical definition, a CRF can be
represented as a factor graph (see Figure 5). By defining
the factors ψ(y) = p(y) and ψ(x, y) = p(x|y) we get an
undirected graph with state and transition probabilities.
p(y|x) =
p(y, x)
p(y, x)
=
=P
0
p(x)
y 0 p(y , x)
min(F P F H1i , F P F H2i ) (3)
i=1
edge cylinder corner plane
edge cylinder corner plane
torus
350
300
250
200
plane
corner
cylinder (r=3cm)
edge
torus (r=3cm)
torus
Ratio of points in one bin (%)
Fig. 5. Graphical representation of a CRF, as a factor graph (left), and as
an undirected graph (right).
FPFH for different geometric scanned surfaces
400
150
p(y|x) =
Histogram Intersection Kernel
100
N
Y
Y
1
ψij (yi , yj , xi , xj )
ψi (yi , xi ) (6)
Z(x)
i=1
(i,j)∈C
where the node potentials are
50
01
The potential functions ψc can be split into edge potentials
ψij and node potentials ψi as follows:
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Bins
Fig. 4. Example of Fast Point Feature Histograms for points lying on
primitive 3D geometric surfaces.
The results show that if the computational parameters are
chosen carefully (i.e. the scale), the features are informative
enough to differentiate between points lying on different
geometric surfaces.
V. C ONDITIONAL R ANDOM F IELDS
Generative graphical models, like Naive Bayes or Hidden
Markov Models, represent a joint probability distribution
p(x, y), where x expresses the observations and y the classification label. Due to the inference problem this approach is
not well applicable for fast labeling of multiple interacting
features like in our case. A solution is to use discriminative
models, such as Conditional Random Fields, which represent
a conditional probability distribution p(y|x). That means
there is no need to model the observations and we can
avoid making potentially erroneous independence assumptions among these features [17]. The probability p(y|x) can
be expressed as:
X
L
ψi (yi , xi ) = exp{ (λL
i xi )yi }
(7)
L
and the edge potentials are
X
ψij (yi , yj , xi , xj ) = exp{ (λij xi xj )yiL yjL }
(8)
L
Learning in a Conditional Random Field is performed
by estimating the node weights λi = {λ1i , . . . , λL
i } and
the edge weights λij = {λ1ij , . . . , λL
ij }. The estimation is
done by maximizing the log-likelihood of p(y|x) [19]. One
way for solving this nonlinear optimization problem is by
applying the Broyden-Fletcher-Goldfarb-Shannon (BFGS)
method. Specially Limited-memory BFGS gains fast results
by approximating the inverse Hessian Matrix.
The given problem of labeling neighbored laser-scanned
points motivates the exploitation of the graphical structure of
CRFs. In our work, each conditioned node represents a laserscanned point and each observation node a calculated feature.
Figure 6 shows a simplified version of our model. The output
nodes are connected with their observation nodes and the
observation nodes of their surrounding neighbors, and each
observation node is connected to its adjacent observation
nodes.
Calculated
Features
x11
x12
x13
x21
x22
x23
TABLE I
F EATURE ESTIMATION , MODEL LEARNING , AND TESTING RESULTS FOR
F IGURE 2. T HE METHOD INDICES REPRESENT THE
PFH (1 ) AND FPFH (2 ). A LL COMPUTATIONS
WERE PERFORMED ON A C ORE 2D UO @ 2 GH Z WITH 4 GB RAM.
THE DATASET IN
TYPE OF FEATURES USED :
Fig. 6.
y1
Feature
Estimation
(pts/s)
Model
Training
(s)
Model
Testing
(pts/s)
Accuracy
CRF1
SVM1
CRF2
SVM2
331.2
331.2
8173.4
8173.4
0.32
1.88
0.091
1.98
18926.17
1807.11
76996.59
4704.90
89.58%
90.13%
97.36%
89.67%
y2
Simplified version of our Conditional Random Field model
VI. D ISCUSSIONS AND E XPERIMENTAL R ESULTS
To validate our framework, we have performed several
experiments of point cloud classification for real world
datasets representing table scenes. The 3D point clouds were
acquired using a SICK LMS400 laser scanner, mounted on
a robot arm and sweeped in front of the table (see Figure 7).
The average acquisition time was ≈1 s per complete 3D point
cloud, and each dataset comprises around 70000 − 80000
points. In total we acquired 10 such datasets with different
objects and different arrangements (see Table II).
Fig. 7. Left: the robot used in our experiments; right: an example of one
of the scanned table scenes.
Error norm and Learning Curve
40000
1.0
FPFHF Error Norm
FPFHF Test Accuracy
PFH Error Norm
PFH Test Accuracy
35000
30000
0.5
15000
0.4
10000
0.3
5000
Fig. 8. Left: raw point cloud dataset; right: the segmentation of the table
(green) and the objects supported by it (random colors) from the rest of the
dataset (black).
Fig. 9.
0.8
0.6
20000
00
0.9
0.7
25000
Error norm
Since we are only interested in classifying the points belonging to the objects on the table, we prefilter the raw point
clouds by segmenting the table surface and retaining only the
individual clusters lying on it. To do this, we assumed that the
objects of interest are lying on the largest (biggest number of
inliers) horizontal planar component in front of the robot, and
we used a robust estimator to fit a plane model to it. Once
the model was obtained, our method estimates the boundaries
of the table and segments all object clusters supported by it
(see Figure 8). Together with the table inliers, this constitutes
the input to our feature estimation and surface classification
experiments, and accounts for approximatively 22000-23000
points (≈ 30% of the original point cloud data).
To learn a good model, we used object clusters from 3 of
the data sets, and then tested the results on the rest. This is in
contrast to our previous formulation in [1] where we trained
the classifiers with synthetically noisified data. The training
set was built by choosing primitive geometric surfaces out of
the data sets, like corners, cylinders, edges, planes, or tori,
both concave and convex. Our initial estimate is that these
primitives account for over 95% data in scenes similar to
ours.
We used the training data to compute two different models,
using Support Vector Machines and Conditional Random
Fields. Since ground truth was unavailable at the data acquisition stage, we manually labeled two scenes to test the
individual accuracy of the two machine learning models
trained. Additionally, on the same datasets, we estimated
and tested two extra SVM and CRF models based on PFH
features. Table I presents the computation time spent for both
the feature estimation, and model training and testing parts.
As shown, the FPFH algorithm easily outperforms the
PFH algorithm in computation performance. It calculates
point histograms over 20 times faster while still gaining
an edge in CRF classification accuracy. The training error
curves for the CRF models for both PFH and FPFH are
shown in Figure 9. Since the error norm becomes very small
after approximatively 50 iterations, we stopped the model
training there. Table II presents visual examples of some
of the classified table scenes with both the SVM and CRF
models.
Accuracy
Labeled
laser points
Method
0.2
10
20
30
40
Training Iterations
50
600.1
Test accuracy and training error curves for the two CRF models.
TABLE II
A SUBSET OF 4 DIFFERENT DATASETS USED TO TEST THE LEARNED MODELS . T HE IMAGES ( TOP TO BOTTOM ) REPRESENT: GRAYSCALE INTENSITY
VALUES , CLASSIFICATION RESULTS OBTAINED WITH THE
Scene 1
SVM MODEL , AND CLASSIFICATION RESULTS OBTAINED WITH THE CRF MODEL .
Scene 2
VII. C ONCLUSIONS AND F UTURE W ORK
In this paper we presented FPFH (Fast Point Feature
Histogram), a novel and discriminative 3D feature descriptor,
useful for the problem of point classification with respect
to the underlying geometric surface primitive the point is
lying on. By carefully defining 3D shape primitives, and
making use of contextual information using probabilistic
graphical models such as CRFs, our framework can robustly
segment point cloud scenes into geometric surface classes.
We validated our approach on multiple table setting datasets
acquired in indoor environments with a laser sensor. The
computational properties of our approach exhibit a favorable
integration for fast, real time 3D classification of laser data.
As future work, we plan to increase the robustness of the
FPFH descriptor against noise to be able to use it for point
classification in datasets acquired using stereo cameras or
other sensors less precised than laser scanners.
Acknowledgments This work was supported by the
CoTeSys (Cognition for Technical Systems) cluster of excellence.
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